Question

A certain substance decays exponentially according to the function n(t) = n0e−ln(3)t where n(t) is the amount of the substance present after t hours and n0 is the initial amount. How long does it take 27 grams of the substance to decay to 9 grams? Simplify your answer completely.

Answer 1

1 hour

Explanation:

Filling in the given information and solving for t, we have ...

9 = 27·e^(-ln(3)t)

9/27 = (1/3)^t . . . . . . . . use e^-ln(3) = 1/3

(1/3)^1 = (1/3)^t . . . . . . . simplify on the left and show an exponent of 1

1 = t . . . . . . . . . . . . . . . match exponents of the same base

It takes 1 hour for 27 grams to decay to 9 grams.

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We have made use of a couple of rules of exponents and logarithms:

a^(bc) = (a^b)^c

e^(ln(x)) = x

-ln(x) = ln(1/x)